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   <h2 class="likechapterHead"><a 
 id="x30-103000"></a>Chapter M&#x00A0;&#x00A0;Matrices</h2>
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</p><!--l. 10--><p class="indent" >   We have made frequent use of matrices for solving systems of equations, and
we have begun to investigate a few of their properties, such as the null space and
nonsingularity. In this chapter, we will take a more systematic approach to the
study of matrices.
</p>
   <div class="likesectionTOCS">
   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-104000">Section MO Matrix Operations</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-105000" id="QQ2-31-105">Subsection MEASM: Matrix Equality, Addition, Scalar Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-106000" id="QQ2-31-106">Subsection VSP: Vector Space Properties</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-107000" id="QQ2-31-107">Subsection TSM: Transposes and Symmetric Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-108000" id="QQ2-31-108">Subsection MCC: Matrices and Complex Conjugation</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-109000" id="QQ2-31-109">Subsection AM: Adjoint of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-110000" id="QQ2-31-110">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-111000" id="QQ2-31-111">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li30.xml#x31-112000" id="QQ2-31-112">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-113000">Section MM Matrix Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-114000" id="QQ2-32-114">Subsection MVP: Matrix-Vector Product</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-115000" id="QQ2-32-115">Subsection MM: Matrix Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-116000" id="QQ2-32-116">Subsection MMEE: Matrix Multiplication, Entry-by-Entry</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-117000" id="QQ2-32-117">Subsection PMM: Properties of Matrix Multiplication</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-118000" id="QQ2-32-118">Subsection HM: Hermitian Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-119000" id="QQ2-32-119">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-120000" id="QQ2-32-120">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li31.xml#x32-121000" id="QQ2-32-121">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-122000">Section MISLE Matrix Inverses and Systems of Linear Equations</a></span>
                                                                          

                                                                          
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-123000" id="QQ2-33-123">Subsection IM: Inverse of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-124000" id="QQ2-33-124">Subsection CIM: Computing the Inverse of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-125000" id="QQ2-33-125">Subsection PMI: Properties of Matrix Inverses</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-126000" id="QQ2-33-126">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-127000" id="QQ2-33-127">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li32.xml#x33-128000" id="QQ2-33-128">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-2.01li33.xml#x34-129000">Section MINM Matrix Inverses and Nonsingular Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li33.xml#x34-130000" id="QQ2-34-130">Subsection NMI: Nonsingular Matrices are Invertible</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li33.xml#x34-131000" id="QQ2-34-131">Subsection UM: Unitary Matrices</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li33.xml#x34-132000" id="QQ2-34-132">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li33.xml#x34-133000" id="QQ2-34-133">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li33.xml#x34-134000" id="QQ2-34-134">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-135000">Section CRS Column and Row Spaces</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-136000" id="QQ2-35-136">Subsection CSSE: Column Spaces and Systems of Equations</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-137000" id="QQ2-35-137">Subsection CSSOC: Column Space Spanned by Original Columns</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-138000" id="QQ2-35-138">Subsection CSNM: Column Space of a Nonsingular Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-139000" id="QQ2-35-139">Subsection RSM: Row Space of a Matrix</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-140000" id="QQ2-35-140">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-141000" id="QQ2-35-141">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li34.xml#x35-142000" id="QQ2-35-142">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-143000">Section FS Four Subsets</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-144000" id="QQ2-36-144">Subsection LNS: Left Null Space</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-145000" id="QQ2-36-145">Subsection CRS: Computing Column Spaces</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-146000" id="QQ2-36-146">Subsection EEF: Extended echelon form</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-147000" id="QQ2-36-147">Subsection FS: Four Subsets</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-148000" id="QQ2-36-148">Subsection READ: Reading Questions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-149000" id="QQ2-36-149">Subsection EXC: Exercises</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-150000" id="QQ2-36-150">Subsection SOL: Solutions</a></span>
<br />   &#x00A0;&#x00A0;<span class="likesubsectionToc" ><a 
href="fcla-xml-2.01li35.xml#x36-151000" id="QQ2-36-151">Annotated Acronyms M: Matrices</a></span>
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